DIMACS REU 2020

Name: Matthew Issac matthewissac (at) rutgers (dot) edu Rutgers University - New Brunswick Algebraic Invariants of Pretzel Knots

Heegaard Floer Homology allows us to associate a knot to a doubly filtered chain complex. Maps between such complexes are called chain maps. We are interested in an involutive chain map, iota_k, and its computation for pretzel knots P(-2,m,n) with m and n odd. This map has been computed for a torus knots and we hope our computation will help provide more examples.

Research Log

Week 1

I was given a brief introduction to knot theory, knot invariants, and some basic homological algebra. Some basic knot invariants I learned about are: three colorability, Alexander polynomial, and Heegaard Floer Homology. I was given a crash course in computing homologies and had plenty of examples to practice with. Moreover, I learned how the Euler Characteristic generalized to bi-graded homologies and how this information encompasses the Alexander polynomial. Finally, I capped the week off by looking into an introductory paper to Heegaard Floer Homology.

Week 2

We were introduced to filtered chain complexes and chain maps. Later on, we saw the Sarkar Involution an involutive, filtered, chain map. This is a crucial piece of data for iota_k, since iota_k^2 is chain homotopic to the Sarkar involution. We also worked through an example of a computation of iota_k for the figure 8 knot.

Week 3

We saw the technical side of CFK^∞, the chain complex we have been working with. We also saw how the generators of this complex come from intersection points on Heegaard diagrams.

Week 4

We have started to read Knot Floer Homology of (1,1)-knots by Goda, Matsuda, and Morifuji. They have already computed CFK^∞ for P(-2,m,n) with m and n odd. We started to understand their notation and worked out what CFK^∞ is for P(-2,5,5) and worked out iota_k for this knot.

Week 5

I began working on a calculator which would accept integers n and m and compute the shape of the complex. There were initally many bugs but I eventually ironed it out and we can now compute more examples.

Week 6

I have begun thinking about a change of basis to do to the complex and how to create a box and staircase pattern.

Week 7

I am beginning to understand how the indices relate to one another, in particular I found certain conditions on when two generators lie in the same filtration level.

Week 8

We have begun discussing the main theorem, in addition to new invariants. We have drafted this and are now preparing our final presentation.

Week 9

Finished our presentation but still more work to be done. We still need to prove main theorem and finish the final change of basis.